A domain is a nonempty connected open set in the ambient space under discussion. In complex analysis this normally means an open connected subset of $\mathbb{C}$. A star-shaped set that is open is a domain, but a domain need not be a simply connected domain and can have a period obstruction to a holomorphic antiderivative.

Let $T:E\to F$ be a bounded linear operator, $E$ a Banach space, and $F$ a normed vector space. Suppose each residual $r$ has an approximate preimage of norm at most $R\Vert r\Vert$ leaving error at most $k\Vert r\Vert$, where $0<k<1$. Correct successive residuals to get vectors $x_j$ with $\Vert x_j\Vert \le R{k}^j\Vert y\Vert$. Their absolutely convergent sum in $E$ maps exactly to $y$, and its norm is at most $R\Vert y\Vert /(1-k)$. Completeness of $F$ is not needed for this argument.

If $H$ is continuous, piecewise twice differentiable, and has derivative jump $[H^{\prime }]=H^{\prime }(y+)-H^{\prime }(y-)$ at $y$, then its distributional second derivative is $H_{\mathrm{reg}}^{\prime \prime }+[H^{\prime }]{\delta }_y$. If the regular part is integrable and the boundary terms vanish, two integrations by parts give the displayed asymptotic expansion by the Riemann-Lebesgue lemma. A jump $[H]$ in $H$ itself instead produces $[H]e^{-iqy}/(iq)$ at leading order. This distinguishes the different decay rates of the streamfunction and streamwise velocity in linear inviscid damping in unbounded Couette flow.

For $0<\alpha ,\beta <1$, this probability generating function has probabilities $p_0=1-\alpha$ and $p_k=\alpha \beta (1-\beta )\cdots (k-1-\beta )/k!$ for $k\ge 1$, where the product after $\beta$ is empty for $k=1$. All probabilities are positive and sum to one, while the expected value is infinite. In a Galton-Watson process starting with one individual, the generation-$n$ probability generating function is $G_n(s)=1-{\alpha }^{(1-{\beta }^n)/(1-\beta )}(1-s{)}^{{\beta }^n}$. The branching extinction probability is $1-{\alpha }^{1/(1-\beta )}$, obtained by taking the limit of $G_n(0)$ and using that extinction by generation $n$ is an increasing sequence of events.

The two-cycle of $g(x)=\mu -x^2$ has points $(1\pm \sqrt{4\mu -3})/2$ and multiplier $4(1-\mu )$. Crossing $\mu =1$ changes its sign and passes through the critical cycle $\{0,1\}$. The sign lift therefore changes from two stable two-cycles to one stable four-cycle. The second iterate is continuous at zero at the transition, but retains jumps at the nonzero preimages of zero. Thus the return-map interpretation concerns a neighborhood of zero, not continuity of the whole second iterate on the real line.

// Target: dynamical-systems.bigb

Antarctic bottom water is a dense southern-origin water mass that ventilates much of the abyssal ocean. Cooling and brine rejection form dense shelf water; its descent and mixing with adjacent waters contribute to Antarctic bottom water production. Shelf production is geographically concentrated and does not mean that every polynya exports water to the abyss.

The Antarctic Circumpolar Current is the eastward ocean circulation encircling Antarctica and connecting the Atlantic, Indian and Pacific oceans. Its direction differs from the predominantly westward Antarctic Coastal Current closer to the continent. Wind stress, oceanic density gradients and seabed geometry shape the current; it is not a single uniform-speed ring.^[ref][ref]

The current illustrates why ocean dynamics require both forcing and boundaries. Atmospheric winds supply momentum, geostrophic balance relates much of the large-scale flow to horizontal pressure gradients, and underwater ridges redirect its path. Measuring the current therefore involves velocity and water-column structure across a section, rather than tracking one surface parcel and assigning that speed to the entire ocean.

The Antarctic Coastal Current is a mainly westward flow near the Antarctic coast. It can transport icebergs along the continental margin; coastal geometry, winds and regional circulation make trajectories variable.

The Antarctic Convergence is the circumpolar oceanic transition where cold Antarctic waters meet and mix with warmer waters farther north. It is associated with the Antarctic Polar Front, a boundary between contrasting water masses rather than a fixed line on a map. The Australian Antarctic Program identifies this mixing region and its enhanced nutrient concentrations.^[ref]

A ship crossing the region can record a marked change in near-surface temperature together with changes in the vertical water-column structure. The physical distinction is between water masses with different density and thermal properties, so a temperature section is more informative than latitude alone. The Antarctic Circumpolar Current follows an eastward path around the continent; the Antarctic Convergence identifies a frontal region within this broader circulation.

For a comoving observer in a spatially flat Friedmann-Lemaître-Robertson-Walker metric, a finite future conformal time interval gives an event horizon at the displayed comoving radius. Its physical radius at time $t$ is $a(t){\chi }_{\mathrm{eh}}(t)$.

For every $ϵ>0$, choose a nonzero smooth spectral bump supported in $(-{ϵ}_0,{ϵ}_0)$ with ${ϵ}_0<\min (ϵ,\pi )$, and let $h$ be its inverse Fourier transform. Then $g$ is a nonzero Schwartz function, vanishes on all integer samples, and its transform is the difference of bumps shifted to $\pi$ and $-\pi$. Its support lies strictly within $[-\pi -ϵ,\pi +ϵ]$. Thus even a small bandwidth increase destroys uniqueness from unit-spaced samples.

Aliasing is the inability to distinguish different continuous frequencies from sampled values. The identity above says that frequencies separated by an integer multiple of the sampling rate $f_s$ give identical samples of a complex Fourier mode. For real cosines, changing the sign of a frequency also gives the same waveform, explaining the $2$-versus-$8$ example at rate $10$. The Nyquist–Shannon sampling theorem prevents this ambiguity within its specified class of bandlimited functions. Before sampling, unwanted higher frequencies must be suppressed if they would violate that assumption; once two waveforms give the same samples, those samples cannot identify which was present.

Integrate the five Lagrange interpolation polynomials at the equally spaced nodes to obtain the displayed rule. It is exact for polynomials through degree four, and symmetry also makes it exact in degree five. Symmetric endpoint, inner-node and midpoint weights $a,b,c$ satisfy $2a+2b+c=2$, $2a+b/2=2/3$, $2a+b/8=2/5$, giving $a=7/45$, $b=32/45$, $c=12/45$.

If $M,N$ are strictly positive traded numeraires and $M/N$ is a mean-one martingale after normalizing its initial value under $Q^N$, this ratio defines $Q^M$. Conditional change of measure then makes every $Q^N$-martingale asset price $S/N$ into the corresponding $Q^M$-martingale $S/M$, when the required integrability holds. In particular a martingale exchange rate under one currency's bank-account measure has a reciprocal martingale under the other currency's measure when their interest rates coincide.

A quadratic number field is a number field of degree of a field extension two, necessarily of the form $\mathbb{Q}(\sqrt{d})$ for a square-free integer $d\ne 0,1$. Its two field embeddings send $a+b\sqrt{d}$ to $a\pm b\sqrt{d}$. If $d>0$, both images lie in the real numbers, giving a totally real number field; if $d<0$, they form a pair related by complex conjugation, giving an imaginary quadratic field. The ring of integers of a quadratic field depends on $d$ modulo four.^[ref]

A number field is totally imaginary if it has no embedding into the real numbers. Its embeddings into the complex numbers occur entirely in conjugate pairs. This is also called totally complex; it does not assert that every element of the field is nonreal.

Column density is the number density integrated along a line of sight. Multiplication by an absorption cross-section gives an optical depth. A small column can result from a small local density, a small path length, or a small fraction of atoms in the absorbing state, so density inference requires these quantities to be constrained.

In the ingoing Eddington-Finkelstein coordinates metric $d{s}^2=-f(r)d{v}^2+2dvdr+r^{2}d{\Omega }^2$, where $f=1-2M/r$, the tangent $U=-E{\partial }_r$ is affine when $E>0$ is constant along each ray. The displayed radial partner obeys $N^2=0$ and $U\cdot N=-1$. Since ${\Gamma }^v{}_{rv}={\Gamma }^a{}_{rr}=0$ and ${\Gamma }^r{}_{rv}=-f^{\prime }/2$, its derivative ${\partial }_r{N}^r=f^{\prime }/(2E)$ cancels the connection term, giving ${\nabla }_{U}N=0$. Its screen is the angular two-plane, with optical tensor ${\hat{B}}_{AB}=-Er{\gamma }_{AB}$, null expansion $-2E/r$, and zero null shear and null twist.

For affine null tangent $U$, choose an auxiliary null vector $N$ with $U\cdot N=-1$ and parallel transport it along the rays. The screen-space projector is $P_{ab}=g_{ab}+U_a{N}_b+N_a{U}_b$. The optical matrix of a null congruence describes evolution of screen-projected deviation vectors. Its trace is the null expansion, its symmetric trace-free part is the null shear, and its antisymmetric part is the null twist. In four dimensions $\hat{B}=\theta P/2+\hat{\sigma }+\hat{\omega }$.

The classical stress-energy tensor of a canonical scalar field has a gradient-product term and a metric-proportional term. Contracting with a real null vector removes the latter, leaving the displayed square for a real field. This proves the null energy condition for a minimally coupled canonical scalar, including the massless case. It is not a statement about arbitrary curvature couplings, wrong-sign kinetic terms or a renormalized quantum stress-energy tensor.